2008 Japan MO Finals

Part of Japan MO Finals

Subcontests

(5)

1

Maximum value of the number of Turn

There are 2008 red cards and 2008 white cards. 2008 players sit down in circular toward the inside of the circle in situation that 2 red cards and 2 white cards from each card are delivered to each person. Each person conducts the following procedure in one turn as follows.

(*)

If you have more than one red card, then you will pass one red card to the left-neighbouring player. If you have no red card, then you will pass one white card to the left -neighbouring player. Find the maximum value of the number of turn required for the state such that all person will have one red card and one white card first.

1

Angle between Line PQ and BC

Given an acute-angled triangle

ABC

with circumcenter

O.

The circle passing through two points

A,\ O

intersects with the line

AB

AC

P,\ Q

A

respectively. If the lengths of the line segments

PQ,\ BC

are equal, then find the angle

\leq 90^\circ

PQ

BC

1

Proof of Existence of n

Can it be existed postive integers

n

such that there are integers

b

and non zero integers a_i\ (i\equal{}1,\ 2,\ \cdots n) for rational numbers

r

which satisfies r\equal{}b\plus{}\sum_{i\equal{}1}^n \frac{1}{a_i}?

1

Proof of P(a^2)\neq 1

P(x)

be a polynomial with integer coefficients such that P(n^2) \equal{} 0 for some non zero integers

n.

P(a^2)\neq 1

for all non zero rational numbers

a\neq 0.

1

f(x+y)f(f(x)-y)=xf(x)-yf(y)

Find all functions

f : \mathbb{R} \to \mathbb{R}

such that f(x\plus{}y)f(f(x)\minus{}y)\equal{}xf(x)\minus{}yf(y) for all

\ x,y \in \mathbb{R}.